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Sunday, 23 October 2016
Chiral graded structures in biological plywoods and in the beetle cuticle
Published Date November 2014, Vol.3:18–22,doi:10.1016/j.colcom.2015.04.001 Open Access, Creative Commons license Author
Oscar F. Aguilar Gutierrez
Alejandro D. Rey,
Department of Chemical Engineering, McGill University, Montreal, Quebec H3A 2B2, Canada
Available online 17 April 2015.
Abstract
Biological chiral fibrous composites, known as biological plywoods, found throughout nature including the exoskeletons of insects and plant cell walls have optimized structural and functional properties, such as the iridescent colors observed in beetle cuticles. In many cases the micron-range chirality of the fibrous ordering is usually spatially graded, multi-periodic or layered as opposed to uniform. The challenge to discover structure–property relations in biological plywoods relies on the accuracy of determining the usually space-dependent chiral pitch of the plywoods. Here we use a recently developed geometric model and computational visualization tool to determine the complex spatial gradients present in beetle cuticle which is a canonical example of graded biological plywoods, extensively studied using optical methods. The proposed computational structural characterization procedure offers a complementary tool to optical and other experimental measurements. The new procedure has wide application in biological material characterization and in biomimetic engineering of structural and functional materials.
Graphical abstract
Keywords
Biological chiral liquid crystals
Helicoidal plywoods
Chirality gradients
Beetle cuticle
Biological materials are distinguished by hierarchical structures, multi-functionality and self-assembly, which are attributes of liquid crystals and unsurprisingly mesophase organization is then found throughout nature with building blocks like collagen, and chitin [1], [2], [3] and [4]. Biological plywoods (BPs) are recognized as solid liquid crystal analogues formed through an efficient entropy-driven self-assembly process [1], [4], [5] and [6]. Bouligand was the first who successfully described these materials with the twisted plywood architecture model (TPA) [7] in which fibrils are arranged as in a chiral nematic liquid crystal, presenting an average orientation of fibrils that change orthogonally to the fibril orientation; the micron-range distance required for a full (2π) rotation is known as the pitch po, whose sign (+ or -) represents the handedness of the helical axis (h); see Fig. 1. TPAs include: i) ideal plywoods with constant pitch (p0 = const) and helix axis, and ii) non-ideal TPAs with pitch gradients (p0(z)). Non-idealities are due to specific cellular processes [7] and a response to external stimuli such as the pH dependence of the self-assembly process observed in collagen [8] and [9] with structures that range from ideal TPA to orthogonal plywoods, with abrupt changes of 90° in the fibril orientation on adjacent planes [10]. Other ubiquitous non-idealities such as defects and gradients in both hand po arise during the plywood self-assembly in the presence of secondary phases [11] which result in poly-domain helicoids [12] and [13].
Fig. 1. (a–b) Schematics of the ideal biological plywoods. Oblique sections (b) give rise to the characteristic arced patterns. (c) Normal view schematic of the helical axis, defined by the helix orientation h and the pitch po.
In this paper we focus on non-idealities arising from spatial pitch variations (dp0(z)/dz≠ 0) in beetle cuticles (‘aurigans scarab’) [14], where the chiral arrangement results in color iridescence [15] and possible thermal regulation [16]; color iridescence due to surface wrinkling is discussed elsewhere [17] and [18]. This cuticle can be classified as a graded plywood with pitch gradients in h direction. For ideal BPs, oblique incisions lead to the ubiquitous arced patterns with identical arcs throughout the incision plane as a result of the homogeneity in both h and po observed by Bouligand in crab cuticles [7], whereas for non-ideal plywood arcs with varying periodicity can be observed in the incision plane due to the pitch variation. The reconstruction of the 3D structure out of 2D observations is a classical inverse problem, and the well-known challenges are due to unavoidable ill-conditioning (small changes in input lead to large changes in output) and solution multiplicity in structure reconstruction.
A method for characterizing ideal plywoods that essentially overcomes these challenges has been reported recently [19] for ideal plywoods using geometric modeling and computational tools, which is extended here to a representative graded plywood of the ‘aurigans scarab’ cuticle. These new tools do not rely on optics as the Cano–Grandjean defect disclination line method or the Bragg reflection, methods that are known to be limited by its range of applicability and/or its accuracy [20] and [21], hence the geometric modeling and computational procedure can complement experimental techniques, leading to a more robust characterization of the ubiquitous chiral structure.
The objective of this communication is to describe a methodology capable of reconstructing the 3D structure of a graded plywood (po(z)) of ‘aurigans scarab’ as a canonical example of a complex biological plywood. In this paper the experimentally determined non-monotonic spatial pitch variations in the beetle cuticle [14] are used to construct arbitrary in silico arced patterns, since no experimental arced patterns are accessible, and then each of these sections are used to perform 3D chiral reconstructions. When slicing a plywood at an arbitrary incision angle (α), we obtain a 2D layered arced patterns of period L, and each arc is characterized by a maximum curvature κmax. We introduce a novel geometry–chirality phase diagram and demonstrate that in this diagram a set of (L, κmax) values found experimentally leads to one and only one value of the pitch regardless the incision angle α. The key aspect of our method is that when the 2D patterns are space dependent (L(z), κmax(z)), the predicted chirality from our model is also space dependent, po(z). Finally we prove that the accuracy of the 3D reconstruction methodology decreases with α.
The analytical description of biological plywoods are obtained through the geometric modeling previously published [19] is summarized as follows: the trajectories of the arced patterns are a function of α and po (implicitly a function of the periodicity of the arcs “L”), leading to a system with one degree of freedom with multiple solutions. To eliminate this degree of freedom, the curvature κ of the arcs, which is a function of αand L, is introduced. By measuring the curvature of the experimentally observed arcs and fitting the analytical expression one can obtain the incision angle α and finally the pitch po with a straightforward calculation. The equations for the arced patterns and the curvature within each layer are [19]:
equation1
equation2a,b
For graded plywoods as the beetle's plywood, the maximum curvature on a given layer is:
equation3
Combining the maximum curvature κmax and the 2D periodicity of the arcs L, leads to the following equation that represents the phase diagram in terms of L and κmax:
equation4
Eq. (4) is a family of hyperbolas for different values of po, shown in Fig. 2. The negative values for curvature indicate a shift in the handedness of the helical axis. Considering one of the hyperbolas for constant pitch, there are an infinite number of combinations of periodicity and inverse of curvature leading to that precise value of pitch. However, there is one and only one value of the pitch that intercepts both axis of the phase plane for a particular value of α as established in Eqs. (3) and (4). The phase diagram (Fig. 2) shows several hyperbolas for different values of po, and the straight lines from Eq. (3) show that for a given α there is only one combination of curvature κmax and 2D periodicity L intercepting these lines. The two limiting lines correspond to α = 0° and α = 90°, corresponding to cuts normal and parallel to the helix. We note that as the helix vector h is usually unknown a priori, finding α through Eq. (3) is a crucial step. Small α angles lead to ill-conditioning as the hyperbolas essentially asymptotes the lines. The practical implementation to determine po(z) using the phase diagram (Fig. 2) is as follows: (1) perform the experimental sectioning of the BP, (2) determine (L, κmax) in a given layer of the 2D arc patterns, (3) use (L, κmax) to determine from the phase diagram (Fig. 2) po(z) and α for the given layer, (4) repeat procedure for all layers, and (5) finally obtain po(z) spatial profile. This procedure is schematically depicted in Fig. 3. The stars shown in such figure indicate only one interception of L, κmax per each α leading to only one value of po.
Fig. 2. Chirality phase diagram in terms of L and arc's reciprocal maximum curvature κmax for several values of pitch. The hyperbolas are from Eq. (4) and the lines from Eq. (3). Performing a section of a real plywoods yield L, κmax and the plot gives po, and α.
Fig. 3. Schematic of the characterization procedure to find an arbitrary pitch profile po (z): (1) perform the experimental sectioning of the BP (α is unknown), (2) determine (L, κmax) in a given layer of the 2D arc patterns, (3) use (L, κmax) to determine from the phase diagram (Fig. 2) po(z) and α for the given layer (star symbols), (4) repeat procedure for all layers, and (5) finally obtain po(z) spatial profile (star symbols). The scarab photo is adapted from [14].
As mentioned, the arced patterns for the beetle cuticle were created using the 3D visualization tool Mayavi, by first introducing the cholesteric director field and then creating an incision plane with the VectorCutPlane module, and using the experimental information of Libby et al. [14] for aurigans scarab, showing the spatial variation of the pitch presented in Figure E6 (see ESI); we emphasize that although we use the experimental pitch as an input to create the arcs only due to the lack of experimental 2D observations. In reality the procedure would start with the slicing of the biological sample giving the experimental arced patterns. It can be seen (Fig. E6) that the non-monotonic pitch is initially close to 0.48 μm with a quick increase up to 0.52 μm, followed by a further decrease, then going into a region of essentially constant pitch (approximately 0.37 μm), and finally showing a smooth increase. Characteristic visualizations of sectioning (α1 = 15° and α2 = 25°) this synthetic aurigans scarab plywood and resulting graded arc patterns are shown in Fig. 4 (a–b for α1 and c–d for α2). Technical visualization details for plywoods are included in the ESI, and for general information about Mayavi the reader is referred [22]. It is observed that initially small arcs appear followed by larger arcs, as expected with pitch variations, regardless the incision angle. The key impact of the incision angle αis the resulting arc periodicity L and the total number of arcs observed, since at a smaller α the incision plane corresponds to a smaller region of the variation of the pitch in the perpendicular coordinate, and hence less information is observed if the size of the sample is kept constant. Three different incision angles were chosen (α1 = 15°, α2 = 25° and α3 = 40°) to demonstrate that the characterization procedure is accurate regardless of the incision angle as shown in Fig. 4 with a sample size constant of 15 μm. To do this, regions in both incision planes were chosen arbitrarily, these arcs were then isolated and the periodicity and maximum curvature were measured, finally the pitch predictions were obtained. The results in Fig. 5 show good agreement with the experimental observations, with a low error (under 10%); we note that information regarding the experimental errors in [14] is not accessible. The accuracy in the determination of the curvature has to be high because the calculation of the incision angle is highly sensitive to changes of both L and κmax. Additionally, it is observed that the lowest angle incision plane (15°) is capable of reconstructing the pitch only up to about 4 μm in the perpendicular direction while the incision at 25° can predict the pitch variations up to about 8 μm in such direction and that of 40° up to 12 μm. Hence, the higher the angle the more information can be extracted from the incision plane. It is also worth noting that in the pitch predictions shown in Fig. 5, the error is higher in the 15° predictions because of the ill-conditioning as stated concerning Eq. (2-b); the relationship between L and po is highly sensitive to small changes α, when α is small also observed in the phase diagram (Fig. 2) with the hyperbolas converging to the asymptote for small α. From an experimental point of view, low angles should be avoided because of two reasons: i) the restriction of the sample size, if the sample is small, little information can be extracted from those observations, and ii) the error in the pitch tends to be larger in these cases because of the nature of the model, as noted previously. Hence, when taking a sample if one finds a low value of α it is advised to take another incision at a higher angle in order to improve accuracy and to extract more information out of this 2D observation.
Fig. 4. Two in-silico arc patterns, created incision planes (a–b) 25° and (c–d) 15° for the plywood of “aurigans scarab”. The arrowed frames show enlarged 2D arcs for clarity. When po(z), the 2D periodicity L varies. Smaller specimen regions can be reconstructed in the case of the lower angle. Larger scale images are provided in ESI for clarity.
Fig. 5. Chirality reconstruction of aurigans scarab from three different incision planes and comparison with experimental data extracted from [14]. Smaller angles cover shorter “z” distances for a given sample size.
In conclusion, we formulated a structural material characterization procedure for biological plywoods that exhibit non-trivial spatial gradients, using geometric modeling and computational visualization. Determining spatial periodicities and curvatures of 2D arc patterns in conjunction with a theoretically formulated plywood phase diagram yields the sought after cholesteric pitch. The sensitivity of the method to sample sectioning was established and larger sectioning angles are preferred. We demonstrated the applicability of the 3D chiral reconstruction method for graded plywoods using experimental data for the “aurigans scarab”. The proposed computational method complements optical characterization tools that must overcome the difficult challenge of chirality reconstruction. The new procedure has wide applications in biological material characterization and in biomimetic engineering of structural and functional materials.
This research was supported by a grant from NSERC. OFAG is grateful for the financial support of CONACyT (scholarship number 313480) and SEP.
A. Boudet, M. Mitov, C. Bourgerette, T. Ondarcuhu, R. Coratger
Glassy cholesteric structure: thickness variation induced by electron radiation in transmission electron microscopy investigated by atomic force microscopy
Ultramicroscopy, Volume 88, Issue 4, 2001, pp. 219–229
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